MIT14_30s09_lec21

MIT14_30s09_lec21 - MIT OpenCourseWare http/ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . n N 1 , 4 under H 25 A 14.30 Introduction to Statistical Methods in Economics Lecture Notes 21 Konrad Menzel May 5, 2009 Constructing Hypothesis Tests If X i has support S X , then the sample X = ( X 1 ,...,X n ) has support S n The critical region of a test is a region C X ⊂ S n of the support of the sample for which we reject the null. X . X The following example illustrates most of the important issues in a standard setting, so you should look at this carefully and make sure that you know how to apply the same steps to similar problems. Example 1 Suppose X 1 ,...,X n are i.i.d. with X i ∼ N ( µ, 4) , and we are interested in testing H 0 : µ = 0 against H A : µ = 1 . Let’s first look at the case n = 2 : X 1 X 2 A o Reject : C x Don't reject 1 1 k k We could design a test which rejects for values of X 1 + X 2 which are ”too large” to be compatible with µ = 0 . We can also represent this rejection region on a line: This representation is much easier to use if n is large, so it’s hard to visualize the rejection region in terms of X 1 ,...,X n directly. However, by condensing the picture from n to a single dimension we may loose the ability of specifying really odd-shaped critical regions, but typically those won’t be interesting for practical purposes anyway. ¯ So in this example, we will base our testing procedure on a test statistic T n ( X 1 ,...,X n ) = X n and reject for large values of T n . How do we choose k ? - we’ll have to trade off the two types of error. Suppose now that n = 25 , and since X i ∼ N ( µ, 4) , 4 N , 25 T n := X ¯ ∼ under H 1 Image by MIT OpenCourseWare. 1 1 n = 4 n = 25 n = 100 α β 1 0 k 2 X 1 + X 2 1 0 X n 4 k Don't reject Don't reject C x : reject C x : reject Now we can calculate the probabilities of type I and type II error k − 0 k ¯ α = P ( X > k | µ = 0) = 1 − Φ = Φ − 2 / 5 2 / 5 k − 1 ¯ β = P ( X ≤ k | µ = 1) = Φ 2 / 5 Therefore, fixing any one of α,β,k determines the other two, and that choice involves a specific tradeoff between the probability of type I and type II error - if we increase k , the significance level α goes down, but so does power 1 − β . Specifically, if we choose k = 5 3 , α ≈ 6 . 7% , and β ≈ 15 . 87% . For different sample sizes, we can graph the trade-off between the probability of type I and type II error through the choice of k as follows: A low value of k would give high power, but also a high significance level, so that increasing k would 2 move us to the left along the frontier....
View Full Document

This document was uploaded on 01/30/2010.

Page1 / 7

MIT14_30s09_lec21 - MIT OpenCourseWare http/ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online